# Economics 326: Input Demands. Ethan Kaplan

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1 Economics 326: Input Demands Ethan Kaplan Octobe 24, 202

2 Outline. Tems 2. Input Demands Tems Labo Poductivity: Output pe unit of labo. Y (K; L) L What is the labo poductivity of the US? Output is ouhgly US\$4.7 tillion. The labo foce is oughly 53 million people. Theefoe, the aggegate labo poductivity fo the US is: \$96; 000

3 Ho is this di eent fom GDP pe capita. The main di eence is that e ae measuing output pe oking peson not output pe peson living in the US. Maginal Poduct: The additional output fom an addition unit = Maginal Poduct of Labo = Maginal Poduct of Capital = Maginal Poduct of Land Declining Maginal Poductivity: We usually assume that the maginal poduct of an input declines ith input usage. Think about fam poduction. We have land, labo and capital. If e stat ith 2 people and 40 aces of land and zeo capital and e then buy a tacto, output ill incease a lot. If e then buy a second tacto, output might incease less than ith one tacto because the second tato ill pobably

4 not be used as often as the st. If e then buy a thid tacto, the incease in output ill be vey lo. (Sho gaph). 2 Input Demands The poduce solves the po t maximization poblem choosing the amount of capital and labo to employ. In doing so, the poduce deives input demands. These ae the analogues of Mashallian Demand in consume theoy. They ae a function of pices of inputs and the pice of output. We assume (fo no) that ms act competitively. This is a behavioal assumption. We assume that ms act as if they have no impact on pice. In many models, poduces may actually have a small impact on pice. Hoeve, they do not take into account

5 thei impact on pice hen they choose ho labo and capital to employ and ho much to poduce. Poduces maximize (choosing K,L): (K; L) = pf (K; L) L K The competitive assumption is that p is not a function of K and L but athe an exogenous paamete. The thee paametes ae p; ; : The to vaiables ae K; L: Thee ae to main aspects of a poduction function: Retuns to Scale: No Analogue on the Consume Side

6 Rate of Technical Substitution: Analogy ith Maginal Rate of Substitution. Rate of technical substition is the ate at hich a m tades o one input fo anothe holding output constant. F (K; L) = Y Totally di eentiating, e get: dk dl = 0 =) dk dl Intepetation: if e incease labo by one unit, then e have to incease capital by (o incease capital output units units) in ode to keep (the maginal poduct of labo) is lage, then you ae gaining a lot of output hen you gain one unit of labo. The ate of technical substitution is de ned (simila to the maginal ate of substitutions) as a positive

7 numbe: RT S = dk dl Example ith Cobb-Douglas: F (K; L) = K L What is the etuns to scale. Lets incease capital and labo by a facto : F (K; L) = (K) (L) = K L = + K L In othe ods if e incease capital and labo by a facto, output goes up by same facto + : This poduction function exhibits constant etuns to scale. Notice that hen = ; then output goes up by popotionally ith inputs - if e incease all factos by ; output also goes up by : F (K; L) = K L

8 Lets ty = 2: What happens if e double puchases of labo and capital: F (2K; 2L) = (2K) (2L) = 2 K 2 L = 2K L No e solve fo labo and capital demand. It is vey simila to the consume side. Fist note that e should check second ode conditions to make sue e have a global maximum. Essentially, the second deivative of the po t function (and thus the poduction function) should be negative. We ill sho this using a simple example ith only one facto of poduction. The second ode condition being satis ed basically is the same as

9 assuming a declining maginal poduct. Then: (L) = pf (L) (L) @ 2 = 2 < 0 2 < 0 Retuning to to factos., e have to st ode conditions - one fo each input (emembe, no Lagange multiplie hee!): = 0 max K;L = p pf (K; L) L K Intuition: the maginal poduct of labo is equal to the eal age (sho gaph) o the maginal evenue poduct is equal to the age () :

10 Similaly, e can deive the analogous fomula fo capital: p = 0 max K;L =) = p pf (K; L) L K Intuition: the maginal poduct of labo is equal to the eal age (sho gaph) o the maginal evenue poduct p is equal to the age () : Fom this, e can compute a fomula fo the RTS (ate of technical substitution): dk dl = p p =

11 In the paticula case of Cobb-Douglas, e get: max K;L pk L = pk L = 0 = pk L = 0 (2) No e solve simultaneously fo K and L: Fist, e e-expess the st ode condtions fom equations () and (2): pk L = (3) pk L = (4) No e divide equation (3) by equation (4): K L = hich e can e-expess as: K = L (5)

12 hich e then plug in to (3) : pk L = p! L L = (6) Re-expessing equation (6), e get: L + = p = p! This leads to a solution fo the input demand function fo labo L (p; ; ) : L (p; ; ) = 2 = p 4 p! 3 5! + + +

13 No e can plug ou solution fo L into (5) : K = L = = p = p p!! + +! +! +! Compaative = p +! + + > 0 geate than zeo if + > =) deceasing etuns to scale

14 @ = + p > 0 geate than zeo if if + > 0 =) deceasing etuns to scale We can also ; ; ; : Suppose e plug K and L back into Y (K; L)? What do e get? Optimal output as a function of output pices and input pices. Y (K (p; ; ) ; L (p; ; )) = Y (p; ; ) This function has a famous name in economics. What is it? If e gaph Y (p; ; ) on p (holding xed and ), hat is it called?

15 In ou paticula Cobb-Douglas example, e get: Y (p; ; ) = 2 6 4p 2 6 4p! +! = p! + + What is the Does that make intuitive sense? What happens ith inceasing and constant etuns to scale? Fo inceasing etuns, the poduction function is convex not concave. A solution to the calculus poblem gives a global minimum.

16 Fo constant etuns, it is slightly moe complicated We ill cove both of these cases soon.

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